A second order ensemble algorithm for computing the Navier-Stokes equations

Abstract

We present an efficient, second order ensemble algorithm for computing the Navier-Stokes equations multiple times with different model parameters. The algorithm is based on the Crank-Nicolson Leap-frog (CNLF) scheme and the ensemble timestepping incorporating the artificial compressibility method and the recent scalar auxiliary variable (SAV) idea for developing unconditionally stable schemes for nonlinear flows. The computation of the velocity and pressure is decoupled in the algorithm resulting in smaller linear systems to be solved at each time step. All realizations of the flow corresponding to different model parameters share the same coefficient matrix so that efficient block solvers can be used to reduce the computational cost. The proposed algorithm is efficient in terms of both computer storage and CPU time. We prove the algorithm is long time stable without any timestep conditions. Ample numerical experiments are performed for various flow problems to validate the efficiency and effectiveness of the algorithm.